High-resolution non-parametric spectral estimation using the Hirschman optimal transform

The traditional Heisenberg-Weyl measure quantifies the joint localization, uncertainty, or concentration of a signal in the phase plane based on a product of energies expressed as signal variances in time and in frequency. Unlike the Heisenberg-Weyl measure, the Hirschman notion of joint uncertainty is based on the entropy rather than the energy [1]. Furthermore, its definition extends naturally from the case of infinitely supported continuous-time signals to the cases of both finitely and infinitely supported discrete-time signals, and, as we noted in [2], the Hirschman optimal transform (HOT) is superior to the discrete Fourier transform (DFT) and discrete cosine transform (DCT) in terms of its ability to separate or resolve two limiting cases of localization in frequency, viz pure tones and additive white noise. In this paper we implement a stationary spectral estimation method using an orthogonal matching pursuit method whose dictionary members are constructed from the combination of HOT-based and DFT atoms (elements) [3] in combination with the interpolating procedure developed in [4]. We call the resulting algorithm the smoothed HOT-DFT periodogram. We compare its performance (in terms of frequency resolution) to Quinn´s smoothed periodogram. In particular, we compare the performance of the HOT-DFT with that of the DFT in resolving two close frequency components in additive white Gaussian noise (AWGN). We find the HOT-DFT to be superior to the DFT in frequency estimation, and ascribe the difference to the HOT´s relationship to entropy.